Efficient computation of highly oscillatory integrals with weak singularities by Gauss-type method

He, Guo; Xiang, Shuhuang; Zhu, Enwen

doi:10.1080/00207160.2014.987761

Cited by 16 publications

(8 citation statements)

References 32 publications

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“…In conclusion, several starting values for the modified moments for forward recursion and Oliver's algorithm or Lozier's algorithm are needed. In addition, the three end moments can be computed by using asymptotic expansion in [14] or the method in [18].…”

Section: Fast Computations Of the Modified Momentsmentioning

confidence: 99%

“…Remark 3. We choose 10 points for the Gauss-type method in [18] to evaluate I(j, k, ω), j = 0, 1, 2, 3, 4 for ω = 2k. While for ω = 2k, we compute them by using the formula (2.37) through Meijer G-function, which can be efficiently computed with the Matlab code MeijerG.m [36].…”

Section: )mentioning

confidence: 99%

“…ν (ωx) dx as a sum of two line integrals (which is a special case of Eq. (20) in [18] with f (x) = 1, b = 1), that is…”

Section: )mentioning

confidence: 99%

“…If f is analytic in a sufficiently large complex region containing [0, 1], a numerical steepest descent method [26] was presented by using complex integration theory. The same idea is also applied to the the computation of the integral b a (x − a) α (b − x) β ln(x − a)f (x)e iωx dx, α, β > −1, based on construction of the Gauss quadrature rule with logarithmic weight function [18]. For the integral 1 0 f (x)x α (1 − x) β J m (ωx) dx, α, β > −1, a Filon-type method based on a special Hermite interpolation polynomial at Clenshaw-Curtis points was introduced in [9].…”

Section: Introductionmentioning

confidence: 99%

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On the Numerical Quadrature of Weakly Singular Oscillatory Integral and its Fast Implementation

Xu¹

2018

Taiwanese J. Math.

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In this paper, we present a Clenshaw-Curtis-Filon-type method for the weakly singular oscillatory integral with Fourier and Hankel kernels. By interpolating the non-oscillatory and nonsingular part of the integrand at (N + 1) Clenshaw-Curtis points, the method can be implemented in O(N log N ) operations. The method requires the accurate computation of modified moments. We first give a method for the derivation of the recurrence relation for the modified moments, which can be applied to the derivation of the recurrence relation for the modified moments corresponding to other type oscillatory integrals. By using recurrence relation, special functions and classic quadrature methods, the modified moments can be computed accurately and efficiently. Then, we present the corresponding error bound in inverse powers of frequencies k and ω for the proposed method. Numerical examples are provided to support the theoretical results and show the efficiency and accuracy of the method.

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Section: Fast Computations Of the Modified Momentsmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

“…ν (ωx) dx as a sum of two line integrals (which is a special case of Eq. (20) in [18] with f (x) = 1, b = 1), that is…”

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Numerical Quadrature of Weakly Singular Oscillatory Integral and its Fast Implementation

Xu¹

2018

Taiwanese J. Math.

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“…A special Gauss-type quadrature, based on the numerical steepest method, has been proposed for the highly oscillatory integrals with algebraic singularities [7,31,32] for linear oscillators, not applied to general oscillators. There is still much work to compute the oscillatory integrals with logarithmic singularities in efficiency and accuracy.…”

Section: Introductionmentioning

confidence: 99%

A Levin method for logarithmically singular oscillatory integrals

Wang,

Xiang

2019

Preprint

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We propose a new stable Levin method to compute oscillatory integrals with logarithmic singularities and without stationary points. To avoid the singularity, we apply the technique of singularity separation and transform the singular ODE into two non-singular ODEs, which can be solved efficiently by the collocation method. Applying the equivalency of the new Levin method for the singular oscillatory integrals and the Filon method when the oscillator is linear, we consider the convergence of the new Levin method. This new method shares the proposition that less error for higher oscillation. Several numerical experiments are presented to validate the efficiency of the proposed method.

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Levin methods for highly oscillatory integrals with singularities

Wang

Xiang

2020

Sci. China Math.

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In this paper, new Levin methods are presented for calculating oscillatory integrals with algebraic and/or logarithmic singularities. To avoid singularity, the technique of singularity separation is applied and then the singular ODE occurring in classic Levin methods is converted into two kinds of non-singular ODEs. The solutions of one can be obtained explicitly, while those of the other can be solved efficiently by collocation methods. The proposed methods can attach arbitrarily high asymptotic orders and also enjoy superalgebraic convergence with respect to the number of collocation points. Several numerical experiments are presented to validate the efficiency of the proposed methods.

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Efficient computation of highly oscillatory integrals with weak singularities by Gauss-type method

Cited by 16 publications

References 32 publications

On the Numerical Quadrature of Weakly Singular Oscillatory Integral and its Fast Implementation

On the Numerical Quadrature of Weakly Singular Oscillatory Integral and its Fast Implementation

A Levin method for logarithmically singular oscillatory integrals

Levin methods for highly oscillatory integrals with singularities

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